The British No 1 practised with the world No 1 on Monday before dismantling Alexander Shevchenko of Russia 6-3 6-2 6-2.
It was just the sort of performance Norrie needed after losing his previous three matches on the hard courts.
And the 16th seed credited Spanish superstar Alcaraz for his improved display.
“We had a really good practice and I think it really set the tone yesterday and I was able to put it into my match today,” he said.
“He’s always enjoying the practice and bringing a lot of flair and excitement to the practice court.
“It was not an easy match. Shevchenko has had a good year so far. I know his game quite well, I’d practiced with him a few times.
“I was able to play longer points and win some of the tough games. It was a nice match to play in the first round to get a lot of rhythm.”
Norrie will face qualifier Yu Hsiou Hsu, the world number 237, in round two after his surprise win over Australian Thanasi Kokkinakis.
Novak Djokovic, meanwhile, will return to the world No 1 spot regardless of whether he wins the tournament this year.
It follows his near-flawless late-night performance at the US Open.
The 36-year-old Serbian, who missed last year’s tournament due to not being vaccinated against Covid, swept past France’s Alexandre Muller 6-0 6-2 6-3.
Flushing Meadows hosted a tribute to former former champion and equality campaigner Billie Jean King, celebrating 50 years since the US Open offered equal prize money to men and women with Michelle Obama making a speech, before Djokovic took to the court at 11pm local time.
He said: “Well, I knew it was going to be a late night for me, a late start of the match.
“Nevertheless, I was excited to go out on the court.
"I didn’t care if I started after midnight because I was looking forward to this moment for few years, to be out on the biggest stadium in our sport, the loudest stadium in our sport, playing night session.
“It was a special night, they had a ceremony. It took longer than I would probably have wanted but was a great joy to be stepping out on the court.
“I think the performance explains how I felt, particularly in the first two sets. It was kind of lights-out tennis really, almost flawless, perfect first set.
“I probably had the answer for every shot he had in his book. Overall I’m very, very pleased with the way I feel, with the way I’m playing.
"Hopefully I can maintain that level. It’s just the beginning of the tournament, but I already like the level of tennis.”
Djokovic will replace reigning champion Alcaraz at the top of the rankings following the US Open.
It will be the seventh time the No 1 spot has changed hands this year.